Answers

If the problem is with the jack then they should be the same as over-the-ear headphones to fix, if the problem is at the earbud or inside then I would really just shell out the 6$ for a new pair... (Try wiggling various parts to see where the problem is)

As an alternative try resetting the phone (home and power for a few seconds), try a normal itunes restore, or last resort a DFU restore. If you have the same problem after doing all this then bushing is right, you probably have a digitizer problem.

Why don't you take it apart and make an iFixit teardown. I'm sure it would be appreciated and in the process you'd be able to have a look at that bad motherboard of yours and see if you could find any sort of obvious faults.

First try hard to remove it using fine pliers if that doesn't work it might be possible to desolder the headphone port from the motherboard and resolder a new one but I don't know where you'd find the jack. This would also be a professional job.

Try holding down the home button while plugging it in this will bring it into DFU mode not Recovery and you might be able to restore. I doubt this problem is harware related as I have seen this problem solved many times by software means. G'luck Liam Speirs

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